Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?tbcon
Estimates the reciprocal of the condition number of a triangular band matrix.
Syntax
call stbcon ( norm , uplo , diag , n , kd , ab , ldab , rcond , work , iwork , info )
call dtbcon ( norm , uplo , diag , n , kd , ab , ldab , rcond , work , iwork , info )
call ctbcon ( norm , uplo , diag , n , kd , ab , ldab , rcond , work , rwork , info )
call ztbcon ( norm , uplo , diag , n , kd , ab , ldab , rcond , work , rwork , info )
call tbcon ( ab , rcond [ , uplo ] [ , diag ] [ , norm ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
stbcon dtbcon ctbcon ztbcon tbcon
The routine estimates the reciprocal of the condition number of a triangular band matrix A in either the 1-norm or infinity-norm:
\(\kappa_{1}(A) = ||A||_{1} ||A^{-1}||_{1} = \kappa_{\infty}(A^{T}) = \kappa_{\infty}(A^{H})\)
\(\kappa_{\infty}(A) = ||A||_{\infty} ||A^{-1}||_{\infty} = \kappa_{1}(A^{T}) = \kappa_{1}(A^{H})\) .
Input Parameters
norm
CHARACTER*1 . Must be ‘1’ or ‘O’ or ‘I’ .
If norm = ‘1’ or ‘O’ , then the routine estimates the condition number of matrix A in 1-norm.
If norm = 'I' , then the routine estimates the condition number of matrix A in infinity-norm.
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ . Indicates whether A is upper or lower triangular:
If uplo = 'U' , the array ap stores the upper triangle of A in packed form.
If uplo = 'L' , the array ap stores the lower triangle of A in packed form.
diag
CHARACTER*1 . Must be ‘N’ or ‘U’ .
If diag = 'N' , then A is not a unit triangular matrix.
If diag = 'U' , then A is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array ab .
n
INTEGER . The order of the matrix A ; n ≥ 0.
kd
INTEGER . The number of superdiagonals or subdiagonals in the matrix A ; kd ≥ 0.
ab , work
REAL for stbcon
DOUBLE PRECISION for dtbcon
COMPLEX for ctbcon
DOUBLE COMPLEX for ztbcon .
The array ab(ldab,*) contains the band matrix A . The second dimension of ab must be at least max(1,n) .
The array work is a workspace for the routine. The dimension of work(*) must be at least max(1, 3*n) for real flavors and max(1, 2*n) for complex flavors.
ldab
INTEGER . The leading dimension of the array ab . ( ldab ≥ kd +1).
iwork
INTEGER . Workspace array, size at least max(1, n) .
rwork
REAL for ctbcon
DOUBLE PRECISION for ztbcon .
Workspace array, size at least max(1, n) .
Output Parameters
rcond
REAL for single precision flavors.
DOUBLE PRECISION for double precision flavors.
An estimate of the reciprocal of the condition number. The routine sets rcond =0 if the estimate underflows; in this case the matrix is singular (to working precision). However, anytime rcond is small compared to 1.0, for the working precision, the matrix may be poorly conditioned or even singular.
info
INTEGER .
If info = 0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
LAPACK 95 Interface Notes
Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions .
Specific details for the routine tbcon interface are as follows:
ab
Holds the array A of size (kd+1,n) .
norm
Must be ‘1’ , ‘O’ , or ‘I’ . The default value is ‘1’ .
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
diag
Must be ‘N’ or ‘U’ . The default value is ‘N’ .
Application Notes
The computed rcond is never less than r (the reciprocal of the true condition number) and in practice is nearly always less than 10 r . A call to this routine involves solving a number of systems of linear equations \(A x = b\) ; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2*n(kd + 1) floating-point operations for real flavors and 8*n(kd + 1) operations for complex flavors.